3.592 \(\int \frac{x^m}{\left (a+b x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=66 \[ \frac{x^{m+1} \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{3}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a (m+1) \sqrt{a+b x^3}} \]

[Out]

(x^(1 + m)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[3/2, (1 + m)/3, (4 + m)/3, -((b
*x^3)/a)])/(a*(1 + m)*Sqrt[a + b*x^3])

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Rubi [A]  time = 0.059729, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x^{m+1} \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{3}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a (m+1) \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a + b*x^3)^(3/2),x]

[Out]

(x^(1 + m)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[3/2, (1 + m)/3, (4 + m)/3, -((b
*x^3)/a)])/(a*(1 + m)*Sqrt[a + b*x^3])

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Rubi in Sympy [A]  time = 7.32391, size = 54, normalized size = 0.82 \[ \frac{x^{m + 1} \sqrt{a + b x^{3}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{a^{2} \sqrt{1 + \frac{b x^{3}}{a}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m/(b*x**3+a)**(3/2),x)

[Out]

x**(m + 1)*sqrt(a + b*x**3)*hyper((3/2, m/3 + 1/3), (m/3 + 4/3,), -b*x**3/a)/(a*
*2*sqrt(1 + b*x**3/a)*(m + 1))

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Mathematica [A]  time = 0.0547945, size = 66, normalized size = 1. \[ \frac{x^{m+1} \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{3}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a (m+1) \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/(a + b*x^3)^(3/2),x]

[Out]

(x^(1 + m)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[3/2, (1 + m)/3, (4 + m)/3, -((b
*x^3)/a)])/(a*(1 + m)*Sqrt[a + b*x^3])

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \[ \int{{x}^{m} \left ( b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(b*x^3+a)^(3/2),x)

[Out]

int(x^m/(b*x^3+a)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(b*x^3 + a)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^m/(b*x^3 + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(b*x^3 + a)^(3/2),x, algorithm="fricas")

[Out]

integral(x^m/(b*x^3 + a)^(3/2), x)

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Sympy [A]  time = 5.5495, size = 53, normalized size = 0.8 \[ \frac{x x^{m} \Gamma \left (\frac{m}{3} + \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{3}{2}} \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m/(b*x**3+a)**(3/2),x)

[Out]

x*x**m*gamma(m/3 + 1/3)*hyper((3/2, m/3 + 1/3), (m/3 + 4/3,), b*x**3*exp_polar(I
*pi)/a)/(3*a**(3/2)*gamma(m/3 + 4/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(b*x^3 + a)^(3/2),x, algorithm="giac")

[Out]

integrate(x^m/(b*x^3 + a)^(3/2), x)